Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfald GIF version

Theorem nfald 1714
 Description: If 𝑥 is not free in 𝜑, it is not free in ∀𝑦𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 6-Jan-2018.)
Hypotheses
Ref Expression
nfald.1 𝑦𝜑
nfald.2 (𝜑 → Ⅎ𝑥𝜓)
Assertion
Ref Expression
nfald (𝜑 → Ⅎ𝑥𝑦𝜓)

Proof of Theorem nfald
StepHypRef Expression
1 nfald.1 . . . 4 𝑦𝜑
21nfri 1480 . . 3 (𝜑 → ∀𝑦𝜑)
3 nfald.2 . . 3 (𝜑 → Ⅎ𝑥𝜓)
42, 3alrimih 1426 . 2 (𝜑 → ∀𝑦𝑥𝜓)
5 nfnf1 1504 . . . 4 𝑥𝑥𝜓
65nfal 1536 . . 3 𝑥𝑦𝑥𝜓
7 hba1 1501 . . . 4 (∀𝑦𝑥𝜓 → ∀𝑦𝑦𝑥𝜓)
8 sp 1469 . . . . 5 (∀𝑦𝑥𝜓 → Ⅎ𝑥𝜓)
98nfrd 1481 . . . 4 (∀𝑦𝑥𝜓 → (𝜓 → ∀𝑥𝜓))
107, 9hbald 1448 . . 3 (∀𝑦𝑥𝜓 → (∀𝑦𝜓 → ∀𝑥𝑦𝜓))
116, 10nfd 1484 . 2 (∀𝑦𝑥𝜓 → Ⅎ𝑥𝑦𝜓)
124, 11syl 14 1 (𝜑 → Ⅎ𝑥𝑦𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1310  Ⅎwnf 1417 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1404  ax-7 1405  ax-gen 1406  ax-4 1468  ax-ial 1495 This theorem depends on definitions:  df-bi 116  df-nf 1418 This theorem is referenced by:  dvelimALT  1959  dvelimfv  1960  nfeudv  1988  nfeqd  2268  nfraldxy  2439  nfiotadxy  5047  nfixpxy  6563  bdsepnft  12768  strcollnft  12865
 Copyright terms: Public domain W3C validator